Re: quantised energy being due to Planck.
I'm not so sure about that. I want to say yes and no. Planck employed quantised energy in his equation but it was Einstein that put forward the radical idea of quantising the elecromagnetic radiation field comparing it to a gas. The notion of a light particle or 'light complex' also appears in section eight of Einstein's paper on the electrodynamics of moving bodies. Planck thought the quantisation was more something to do with the interaction of the radiation with the matter of the walls of the blackbody and wasn't keen about Einstein's idea.

My professor explained that Planck was forced to make the (at the time seemingly random and non-sensical) assumption that energy was quantised in order to solve a problem in blackbody radiation, but that Einstein was the first to begin to make sense of the result. Quantum Mechanics was then further developed/fleshed out by buys like Bohr, Heisenberg, Fermi, Dirac, Pauli etc.

Thank this one to Maxwell, Einstein was able to make it a postulate of Relativity because it was a well known property.

Actually, the constancy of the speed of light wasn't a well known property at the time Einstein formulated SR. At the time, it was an educated guess based on some indirect experimental evidence such as the Michelson Morley experiment. It by no means was a well known property and at the time many physicists did not believe it to be true.